Stone-Geary utility function, derivation of Marshallian demand

Question

I am reading a paper on structural change. It has three sectors and it features non homothetic utility function, namely a CES with some thresholds for the consumption of the three goods. The UMP is as follows: $$ \max_{C_k; k=1,2,3} \left(\sum_{k}\omega^{\frac{1}{\epsilon}}_k\left(C_k-\bar{C}_k\right)^{\frac{\epsilon-1}{\epsilon}}\right)^\frac{\epsilon}{\epsilon-1} $$ Subject to the budget constraint $$ \sum_{K}\left(P_k C_k\right)=W $$ with $\omega_k$ summing up to 1. What I find online is mainly a derivation of the log-utility case, which is not that suitable for my case. Also, in the paper authors use a price index which is really close to the one of the Dixit-Stiglitz case.

I'm currently stuck at the level of the FOC, and I wonder what are the steps to derive a proper demand function from these premises. Can anyone show me how to find this demand function?

Answer 1

If I understand your question correctly, you can take the log of the original utility function to simplify the calculations. The demand derived will be the same as that which is derived from the original utility function. This follows since the log function is a monotone function, and preferences are preserved under monotonic transformations of utility. Thus, the "log-utility" case that you find online is exactly what you want to work with since it would simplify things quite a bit.